Abstract
Given an infinite word \(\ensuremath{\mathbf{x}} \) over an alphabet A, a letter b occurring in \(\ensuremath{\mathbf{x}} \), and a total order σ on A, we call the smallest word with respect to σ starting with b in the shift orbit closure \(\ensuremath{\mathcal{S}} _{\ensuremath{\mathbf{x}} }\) of \(\ensuremath{\mathbf{x}} \) an extremal word of \(\ensuremath{\mathbf{x}} \). In this paper we consider the extremal words of morphic words. If \(\ensuremath{\mathbf{x}} = g(f^{\omega}(a))\) for some morphisms f and g, we give a simple condition on f and g that guarantees that all extremal words are morphic. An application of this condition shows that all extremal words of binary pure morphic words are morphic. Our technique also yields easy characterizations of extremal words of the Period-doubling and Chacon words and a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin-Shapiro word.
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Currie, J.D., Rampersad, N., Saari, K. (2013). Extremal Words in the Shift Orbit Closure of a Morphic Sequence. In: Béal, MP., Carton, O. (eds) Developments in Language Theory. DLT 2013. Lecture Notes in Computer Science, vol 7907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38771-5_14
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DOI: https://doi.org/10.1007/978-3-642-38771-5_14
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